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arxiv: 2606.29882 · v1 · pith:NO3K3IAYnew · submitted 2026-06-29 · 🧮 math.GT · math.DG

Monopole triangle over integers

Haochen Qiu , Fan Ye This is my paper

Pith reviewed 2026-06-30 04:02 UTC · model grok-4.3

classification 🧮 math.GT math.DG
keywords monopole Floer homologysurgery exact triangleSeiberg-Witten Floer homologyinteger coefficientsKhovanov homologyspectral sequencebranched covers
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The pith

The surgery exact triangle for monopole Floer homology holds over integer coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the surgery exact triangle relating the monopole Floer homologies of three manifolds obtained by different surgeries on a knot is valid when the coefficients are the integers. This extends versions previously established only for coefficients modulo 2 or over the rationals. The result yields a spectral sequence from the odd Khovanov homology of an oriented link in the three-sphere to the monopole Floer homology of its double branched cover, again over the integers. A sympathetic reader would care because the integer-coefficient version supplies finer information about torsion in these groups and confirms a conjecture on the existence of the spectral sequence.

Core claim

The central claim is that a modification of the local system on monopole Floer homology, together with an adaptation of an earlier computation, produces a well-defined exact triangle over the integers without introducing uncontrolled torsion or losing exactness. As a direct consequence, the spectral sequence from odd Khovanov homology of a link to the Floer homology of its double branched cover is obtained over Z.

What carries the argument

A modified local system on monopole Floer homology combined with an adapted computation that carries the exactness of the surgery triangle over the integers.

If this is right

  • The spectral sequence from odd Khovanov homology of an oriented link to monopole Floer homology of its double branched cover holds over the integers.
  • Integer-coefficient versions of the invariants become available for applications that previously required working modulo 2 or over the rationals.
  • The exact triangle can be used to relate the Floer homologies of manifolds obtained by surgeries on the same knot, now with integral coefficients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result may allow direct comparison of torsion subgroups across different coefficient rings in Floer homology.
  • Similar modifications could be tested on other variants of Floer homology to obtain integral exact triangles there as well.
  • Concrete examples of links can now be checked to see whether the spectral sequence detects new distinctions invisible over the rationals.

Load-bearing premise

The modified local system and adapted computation together produce an exact triangle over the integers that does not introduce uncontrolled torsion.

What would settle it

A specific knot for which the three groups in the surgery triangle fail to satisfy the exactness relation when computed with integer coefficients.

Figures

Figures reproduced from arXiv: 2606.29882 by Fan Ye, Haochen Qiu.

Figure 1
Figure 1. Figure 1: The triple composite V1 (cf. [KMOS07, [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: This figure is a low-dimensional description for E1 Y E2 and its neighborhood. Spheres E1 and E2 are depicted by black. They intersect at one point. R1 is depicted by the colorful part. ‚ R1. The boundary of the neighborhood of E1 Y E2. Note that R1 – S 1 ˆ S 2 (see [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Kirby diagram of V1 0 F E1 ` E2 0 ´1 E2 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kirby diagram compatible with stretching S2 E2 E1 ` E2 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: This figure is obtained from [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
read the original abstract

We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsv\'{a}th--Szab\'{o} over $\mathbb{Z}/2$, Lin--Ruberman--Saveliev over $\mathbb{Q}$, and Freeman over $\mathbb{Z}[\sqrt{-1}]$. Our proof is based on a modification of Kronheimer--Mrowka's local system on monopole Floer homology and an adaptation of Freeman's computation. As a standard application, following Bloom and Scaduto, we obtain a spectral sequence $\widetilde{Kh}_{\mathrm{odd}}(L)\Rightarrow \widetilde{HM}_\bullet(-\Sigma_2(L))$ over integer coefficients for an oriented link $L\subset S^3$, thereby solving Ozsv\'{a}th--Rasmussen--Szab\'{o}'s conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove the surgery exact triangle for monopole (Seiberg-Witten) Floer homology over integer coefficients. The argument modifies Kronheimer-Mrowka local systems on monopole Floer homology and adapts Freeman's computation (previously over Z[i]). As an application, following Bloom-Scaduto, it produces a spectral sequence from odd Khovanov homology of a link L to the monopole Floer homology of its double branched cover, resolving the Ozsváth-Rasmussen-Szabó conjecture over Z.

Significance. If the central claim holds, the result would extend the surgery triangle from Z/2, Q, and Z[i] coefficients to Z, enabling integral-coefficient applications in low-dimensional topology and strengthening spectral-sequence arguments. The approach via local-system modification is a natural extension of existing techniques in the field.

major comments (1)
  1. The abstract asserts a complete proof via local-system modification and adaptation of prior work, but the provided text supplies no derivation steps, no verification that the modified local system preserves exactness over Z, and no control on torsion or loss of exactness. This directly impacts the load-bearing claim that the triangle is well-defined over Z (reader's weakest assumption).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for greater explicitness in the proof. We address the major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: The abstract asserts a complete proof via local-system modification and adaptation of prior work, but the provided text supplies no derivation steps, no verification that the modified local system preserves exactness over Z, and no control on torsion or loss of exactness. This directly impacts the load-bearing claim that the triangle is well-defined over Z (reader's weakest assumption).

    Authors: We agree that the current manuscript presents the overall strategy at a high level but does not include sufficient step-by-step derivations or explicit checks that the modified Kronheimer-Mrowka local system preserves exactness over Z, nor does it provide detailed control on torsion. In the revised version we will expand the relevant sections (particularly those adapting Freeman's computation) to supply these verifications, including explicit chain-level arguments and torsion estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation extends the surgery exact triangle to integer coefficients via a modification of the Kronheimer-Mrowka local system (external citation) and adaptation of Freeman's computation (external citation). No self-citations appear in the provided abstract or description, and the central claim does not reduce any result, prediction, or uniqueness statement to a definition, fit, or ansatz internal to the paper. The argument is presented as building on independent prior constructions without load-bearing self-reference or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the argument rests on standard background results in Seiberg-Witten theory and Floer homology together with a modification of an existing local system; no new free parameters or invented entities are indicated.

axioms (1)
  • domain assumption Standard properties of Seiberg-Witten monopoles and the associated Floer homology chain complexes extend to a modified local coefficient system over Z.
    Invoked to obtain the exact triangle over integers.

pith-pipeline@v0.9.1-grok · 5680 in / 1148 out tokens · 47076 ms · 2026-06-30T04:02:38.432651+00:00 · methodology

discussion (0)

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Reference graph

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13 extracted references · 9 canonical work pages · 2 internal anchors

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